mirror of https://github.com/nmvdw/HITs-Examples
Work on finite sets
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Niels
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FSet.v
672
FSet.v
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@ -1,34 +1,35 @@
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Require Import HoTT.
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Require Import HoTT.
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Require Export HoTT.
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Require Export HoTT.
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Require Import FunextAxiom.
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Module Export FinSet.
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Module Export FinSet.
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Set Implicit Arguments.
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Section FSet.
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Variable A : Type.
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Parameter A: Type.
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Private Inductive FSet : Type :=
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Private Inductive FSet : Type :=
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| empty : FSet
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| E : FSet
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| L : A -> FSet
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| L : A -> FSet
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| U : FSet -> FSet -> FSet.
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| U : FSet -> FSet -> FSet.
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Infix "∪" := U (at level 8, right associativity).
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Notation "∅" := empty.
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Notation "{| x |}" := (L x).
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Infix "∪" := U (at level 8, right associativity).
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Notation "∅" := E.
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Axiom assoc : forall (x y z : FSet),
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Axiom assoc : forall (x y z : FSet ),
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x ∪ (y ∪ z) = (x ∪ y) ∪ z.
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x ∪ (y ∪ z) = (x ∪ y) ∪ z.
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Axiom comm : forall (x y : FSet),
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Axiom comm : forall (x y : FSet),
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x ∪ y = y ∪ x.
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x ∪ y = y ∪ x.
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Axiom neutl : forall (x : FSet),
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Axiom nl : forall (x : FSet),
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∅ ∪ x = x.
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∅ ∪ x = x.
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Axiom neutr : forall (x : FSet),
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Axiom nr : forall (x : FSet),
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x ∪ ∅ = x.
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x ∪ ∅ = x.
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Axiom idem : forall (x : A),
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Axiom idem : forall (x : A),
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(L x) ∪ (L x) = L x.
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{| x |} ∪ {|x|} = {|x|}.
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Axiom trunc : IsHSet FSet.
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Axiom trunc : IsHSet FSet.
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@ -47,14 +48,13 @@ Fixpoint FSet_rec
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{struct x}
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{struct x}
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: P
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: P
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:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P with
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:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P with
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| empty => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => e
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| E => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => e
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| L a => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => l a
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| L a => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => l a
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| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => u
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| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => u
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(FSet_rec H e l u assocP commP nlP nrP idemP y)
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(FSet_rec P H e l u assocP commP nlP nrP idemP y)
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(FSet_rec H e l u assocP commP nlP nrP idemP z)
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(FSet_rec P H e l u assocP commP nlP nrP idemP z)
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end) assocP commP nlP nrP idemP H.
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end) assocP commP nlP nrP idemP H.
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Axiom FSet_rec_beta_assoc : forall
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Axiom FSet_rec_beta_assoc : forall
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(P : Type)
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(P : Type)
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(H: IsHSet P)
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(H: IsHSet P)
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@ -67,11 +67,11 @@ Axiom FSet_rec_beta_assoc : forall
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(nrP : forall (x : P), u x e = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x y z : FSet),
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(x y z : FSet),
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ap (FSet_rec H e l u assocP commP nlP nrP idemP) (assoc x y z)
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (assoc x y z)
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=
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=
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(assocP (FSet_rec H e l u assocP commP nlP nrP idemP x)
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(assocP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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(FSet_rec H e l u assocP commP nlP nrP idemP y)
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(FSet_rec P H e l u assocP commP nlP nrP idemP y)
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(FSet_rec H e l u assocP commP nlP nrP idemP z)
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(FSet_rec P H e l u assocP commP nlP nrP idemP z)
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).
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).
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Axiom FSet_rec_beta_comm : forall
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Axiom FSet_rec_beta_comm : forall
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@ -86,10 +86,10 @@ Axiom FSet_rec_beta_comm : forall
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(nrP : forall (x : P), u x e = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x y : FSet),
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(x y : FSet),
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ap (FSet_rec H e l u assocP commP nlP nrP idemP) (comm x y)
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (comm x y)
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=
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=
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(commP (FSet_rec H e l u assocP commP nlP nrP idemP x)
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(commP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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(FSet_rec H e l u assocP commP nlP nrP idemP y)
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(FSet_rec P H e l u assocP commP nlP nrP idemP y)
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).
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).
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Axiom FSet_rec_beta_nl : forall
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Axiom FSet_rec_beta_nl : forall
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@ -104,9 +104,9 @@ Axiom FSet_rec_beta_nl : forall
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(nrP : forall (x : P), u x e = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : FSet),
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(x : FSet),
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ap (FSet_rec H e l u assocP commP nlP nrP idemP) (neutl x)
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (nl x)
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=
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=
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(nlP (FSet_rec H e l u assocP commP nlP nrP idemP x)
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(nlP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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).
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).
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Axiom FSet_rec_beta_nr : forall
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Axiom FSet_rec_beta_nr : forall
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@ -121,9 +121,9 @@ Axiom FSet_rec_beta_nr : forall
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(nrP : forall (x : P), u x e = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : FSet),
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(x : FSet),
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ap (FSet_rec H e l u assocP commP nlP nrP idemP) (neutr x)
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (nr x)
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=
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=
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(nrP (FSet_rec H e l u assocP commP nlP nrP idemP x)
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(nrP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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).
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).
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Axiom FSet_rec_beta_idem : forall
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Axiom FSet_rec_beta_idem : forall
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@ -138,14 +138,16 @@ Axiom FSet_rec_beta_idem : forall
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(nrP : forall (x : P), u x e = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : A),
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(x : A),
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ap (FSet_rec H e l u assocP commP nlP nrP idemP) (idem x)
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (idem x)
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=
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=
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idemP x.
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idemP x.
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(* Induction principle *)
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Fixpoint FSet_ind
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Fixpoint FSet_ind
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(P : FSet -> Type)
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(P : FSet -> Type)
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(H : forall a : FSet, IsHSet (P a))
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(H : forall a : FSet, IsHSet (P a))
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(eP : P empty)
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(eP : P E)
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(lP : forall a: A, P (L a))
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet), P x -> P y -> P (U x y))
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(uP : forall (x y: FSet), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet)
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(assocP : forall (x y z : FSet)
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@ -158,16 +160,16 @@ Fixpoint FSet_ind
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comm x y #
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comm x y #
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uP x y px py = uP y x py px)
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet) (px: P x),
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(nlP : forall (x : FSet) (px: P x),
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neutl x # uP empty x eP px = px)
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nl x # uP E x eP px = px)
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(nrP : forall (x : FSet) (px: P x),
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(nrP : forall (x : FSet) (px: P x),
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neutr x # uP x empty px eP = px)
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nr x # uP x E px eP = px)
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(idemP : forall (x : A),
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x : FSet)
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(x : FSet)
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{struct x}
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{struct x}
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: P x
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: P x
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:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P x with
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:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P x with
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| empty => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => eP
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| E => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => eP
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| L a => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => lP a
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| L a => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => lP a
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| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => uP y z
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| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => uP y z
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP y)
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP y)
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@ -176,517 +178,279 @@ Fixpoint FSet_ind
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Axiom FSet_ind_beta_assoc : forall
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Axiom FSet_ind_beta_assoc : forall
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(A : Type)
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(P : FSet -> Type)
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(P : FSet A -> Type)
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(H : forall a : FSet, IsHSet (P a))
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(eP : P (empty A))
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(eP : P E)
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(lP : forall a: A, P (L a))
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
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(uP : forall (x y: FSet), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet A)
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(assocP : forall (x y z : FSet)
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(px: P x) (py: P y) (pz: P z),
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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(uP x (U y z) px (uP y z py pz))
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=
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=
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(uP (U x y) z (uP x y px py) pz))
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y: FSet A) (px: P x) (py: P y),
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(commP : forall (x y: FSet) (px: P x) (py: P y),
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comm x y #
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comm x y #
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uP x y px py = uP y x py px)
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet A) (px: P x),
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(nlP : forall (x : FSet) (px: P x),
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nl x # uP (empty A) x eP px = px)
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nl x # uP E x eP px = px)
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(nrP : forall (x : FSet A) (px: P x),
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(nrP : forall (x : FSet) (px: P x),
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nr x # uP x (empty A) px eP = px)
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nr x # uP x E px eP = px)
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(idemP : forall (x : A),
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x y z : FSet A),
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(x y z : FSet),
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apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP)
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apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP)
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(assoc x y z)
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(assoc x y z)
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=
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=
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(assocP x y z
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(assocP x y z
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(FSet_ind P eP lP uP assocP commP nlP nrP idemP x)
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
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(FSet_ind P eP lP uP assocP commP nlP nrP idemP y)
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP y)
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(FSet_ind P eP lP uP assocP commP nlP nrP idemP z)
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP z)
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).
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).
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Axiom FSet_ind_beta_comm : forall
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Axiom FSet_ind_beta_comm : forall
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(A : Type)
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(P : FSet -> Type)
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(P : FSet A -> Type)
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(H : forall a : FSet, IsHSet (P a))
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(eP : P (empty A))
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(eP : P E)
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(lP : forall a: A, P (L a))
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
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(uP : forall (x y: FSet), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet A)
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(assocP : forall (x y z : FSet)
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(px: P x) (py: P y) (pz: P z),
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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(uP x (U y z) px (uP y z py pz))
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=
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=
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(uP (U x y) z (uP x y px py) pz))
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y: FSet A) (px: P x) (py: P y),
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(commP : forall (x y : FSet) (px: P x) (py: P y),
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comm x y #
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comm x y #
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uP x y px py = uP y x py px)
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet A) (px: P x),
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(nlP : forall (x : FSet) (px: P x),
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nl x # uP (empty A) x eP px = px)
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nl x # uP E x eP px = px)
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(nrP : forall (x : FSet A) (px: P x),
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(nrP : forall (x : FSet) (px: P x),
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nr x # uP x (empty A) px eP = px)
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nr x # uP x E px eP = px)
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(idemP : forall (x : A),
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x y : FSet A),
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(x y : FSet),
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apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP) (comm x y)
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apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (comm x y)
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=
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=
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(commP x y
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(commP x y
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(FSet_ind P eP lP uP assocP commP nlP nrP idemP x)
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
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(FSet_ind P eP lP uP assocP commP nlP nrP idemP y)
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP y)
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).
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).
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Axiom FSet_ind_beta_nl : forall
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Axiom FSet_ind_beta_nl : forall
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(A : Type)
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(P : FSet -> Type)
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(P : FSet A -> Type)
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(H : forall a : FSet, IsHSet (P a))
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(eP : P (empty A))
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(eP : P E)
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(lP : forall a: A, P (L a))
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
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(uP : forall (x y: FSet), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet A)
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(assocP : forall (x y z : FSet)
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(px: P x) (py: P y) (pz: P z),
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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(uP x (U y z) px (uP y z py pz))
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=
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=
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(uP (U x y) z (uP x y px py) pz))
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(uP (U x y) z (uP x y px py) pz))
|
||||||
(commP : forall (x y: FSet A) (px: P x) (py: P y),
|
(commP : forall (x y : FSet) (px: P x) (py: P y),
|
||||||
comm x y #
|
comm x y #
|
||||||
uP x y px py = uP y x py px)
|
uP x y px py = uP y x py px)
|
||||||
(nlP : forall (x : FSet A) (px: P x),
|
(nlP : forall (x : FSet) (px: P x),
|
||||||
nl x # uP (empty A) x eP px = px)
|
nl x # uP E x eP px = px)
|
||||||
(nrP : forall (x : FSet A) (px: P x),
|
(nrP : forall (x : FSet) (px: P x),
|
||||||
nr x # uP x (empty A) px eP = px)
|
nr x # uP x E px eP = px)
|
||||||
(idemP : forall (x : A),
|
(idemP : forall (x : A),
|
||||||
idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
||||||
(x : FSet A),
|
(x : FSet),
|
||||||
apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP) (nl x)
|
apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (nl x)
|
||||||
=
|
=
|
||||||
(nlP x (FSet_ind P eP lP uP assocP commP nlP nrP idemP x)
|
(nlP x (FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
|
||||||
).
|
).
|
||||||
|
|
||||||
Axiom FSet_ind_beta_nr : forall
|
Axiom FSet_ind_beta_nr : forall
|
||||||
(A : Type)
|
(P : FSet -> Type)
|
||||||
(P : FSet A -> Type)
|
(H : forall a : FSet, IsHSet (P a))
|
||||||
(eP : P (empty A))
|
(eP : P E)
|
||||||
(lP : forall a: A, P (L a))
|
(lP : forall a: A, P (L a))
|
||||||
(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
|
(uP : forall (x y: FSet), P x -> P y -> P (U x y))
|
||||||
(assocP : forall (x y z : FSet A)
|
(assocP : forall (x y z : FSet)
|
||||||
(px: P x) (py: P y) (pz: P z),
|
(px: P x) (py: P y) (pz: P z),
|
||||||
assoc x y z #
|
assoc x y z #
|
||||||
(uP x (U y z) px (uP y z py pz))
|
(uP x (U y z) px (uP y z py pz))
|
||||||
=
|
=
|
||||||
(uP (U x y) z (uP x y px py) pz))
|
(uP (U x y) z (uP x y px py) pz))
|
||||||
(commP : forall (x y: FSet A) (px: P x) (py: P y),
|
(commP : forall (x y : FSet) (px: P x) (py: P y),
|
||||||
comm x y #
|
comm x y #
|
||||||
uP x y px py = uP y x py px)
|
uP x y px py = uP y x py px)
|
||||||
(nlP : forall (x : FSet A) (px: P x),
|
(nlP : forall (x : FSet) (px: P x),
|
||||||
nl x # uP (empty A) x eP px = px)
|
nl x # uP E x eP px = px)
|
||||||
(nrP : forall (x : FSet A) (px: P x),
|
(nrP : forall (x : FSet) (px: P x),
|
||||||
nr x # uP x (empty A) px eP = px)
|
nr x # uP x E px eP = px)
|
||||||
(idemP : forall (x : A),
|
(idemP : forall (x : A),
|
||||||
idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
||||||
(x : FSet A),
|
(x : FSet),
|
||||||
apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP) (nr x)
|
apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (nr x)
|
||||||
=
|
=
|
||||||
(nrP x (FSet_ind P eP lP uP assocP commP nlP nrP idemP x)
|
(nrP x (FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
|
||||||
).
|
).
|
||||||
|
|
||||||
Axiom FSet_ind_beta_idem : forall
|
Axiom FSet_ind_beta_idem : forall
|
||||||
(A : Type)
|
(P : FSet -> Type)
|
||||||
(P : FSet A -> Type)
|
(H : forall a : FSet, IsHSet (P a))
|
||||||
(eP : P (empty A))
|
(eP : P E)
|
||||||
(lP : forall a: A, P (L a))
|
(lP : forall a: A, P (L a))
|
||||||
(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
|
(uP : forall (x y: FSet), P x -> P y -> P (U x y))
|
||||||
(assocP : forall (x y z : FSet A)
|
(assocP : forall (x y z : FSet)
|
||||||
(px: P x) (py: P y) (pz: P z),
|
(px: P x) (py: P y) (pz: P z),
|
||||||
assoc x y z #
|
assoc x y z #
|
||||||
(uP x (U y z) px (uP y z py pz))
|
(uP x (U y z) px (uP y z py pz))
|
||||||
=
|
=
|
||||||
(uP (U x y) z (uP x y px py) pz))
|
(uP (U x y) z (uP x y px py) pz))
|
||||||
(commP : forall (x y: FSet A) (px: P x) (py: P y),
|
(commP : forall (x y : FSet) (px: P x) (py: P y),
|
||||||
comm x y #
|
comm x y #
|
||||||
uP x y px py = uP y x py px)
|
uP x y px py = uP y x py px)
|
||||||
(nlP : forall (x : FSet A) (px: P x),
|
(nlP : forall (x : FSet) (px: P x),
|
||||||
nl x # uP (empty A) x eP px = px)
|
nl x # uP E x eP px = px)
|
||||||
(nrP : forall (x : FSet A) (px: P x),
|
(nrP : forall (x : FSet) (px: P x),
|
||||||
nr x # uP x (empty A) px eP = px)
|
nr x # uP x E px eP = px)
|
||||||
(idemP : forall (x : A),
|
(idemP : forall (x : A),
|
||||||
idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
||||||
(x : A),
|
(x : A),
|
||||||
apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP) (idem x)
|
apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (idem x)
|
||||||
=
|
=
|
||||||
idemP x.
|
idemP x.
|
||||||
|
|
||||||
Parameter A: Type.
|
End FSet.
|
||||||
|
|
||||||
Theorem idemSet :
|
Parameter A : Type.
|
||||||
forall x: FSet A, U x x = x.
|
Parameter eq : A -> A -> Bool.
|
||||||
Proof.
|
Parameter eq_refl: forall a:A, eq a a = true.
|
||||||
simple refine (FSet_ind _ _ _ _ _ _ _ _ _).
|
|
||||||
- cbn.
|
|
||||||
apply nl.
|
|
||||||
- cbn.
|
|
||||||
apply idem.
|
|
||||||
- cbn.
|
|
||||||
intros.
|
|
||||||
rewrite assoc.
|
|
||||||
rewrite (comm (U x y)).
|
|
||||||
rewrite assoc.
|
|
||||||
rewrite X.
|
|
||||||
rewrite <- assoc.
|
|
||||||
rewrite X0.
|
|
||||||
reflexivity.
|
|
||||||
- cbn.
|
|
||||||
|
|
||||||
(* todo optimisation *)
|
Arguments E {_}.
|
||||||
Theorem FSetRec (A : Type)
|
Arguments U {_} _ _.
|
||||||
(P : Type)
|
Arguments L {_} _.
|
||||||
(e : P)
|
|
||||||
(l : A -> P)
|
Definition isIn : A -> FSet A -> Bool.
|
||||||
(u : P -> P -> P)
|
|
||||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
|
||||||
(commP : forall (x y : P), u x y = u y x)
|
|
||||||
(nlP : forall (x : P), u e x = x)
|
|
||||||
(nrP : forall (x : P), u x e = x)
|
|
||||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
|
||||||
(x : FSet A) : P.
|
|
||||||
Proof.
|
Proof.
|
||||||
simple refine (FSet_ind _ _ _ _ _ _ _ _ _ x).
|
intros a.
|
||||||
- apply e.
|
simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
|
||||||
- apply l.
|
- exact false.
|
||||||
- apply (fun _ => fun _ => u).
|
- intro a'. apply (eq a a').
|
||||||
- cbn.
|
- apply orb.
|
||||||
intros.
|
- intros x y z. destruct x; reflexivity.
|
||||||
transitivity (u px (u py pz)).
|
- intros x y. destruct x, y; reflexivity.
|
||||||
apply transport_const.
|
- intros x. reflexivity.
|
||||||
apply assocP.
|
- intros x. destruct x; reflexivity.
|
||||||
- cbn.
|
- intros a'. destruct (eq a a'); reflexivity.
|
||||||
intros.
|
|
||||||
transitivity (u px py).
|
|
||||||
apply transport_const.
|
|
||||||
apply commP.
|
|
||||||
- cbn.
|
|
||||||
intros.
|
|
||||||
transitivity (u e px).
|
|
||||||
apply transport_const.
|
|
||||||
apply nlP.
|
|
||||||
- cbn.
|
|
||||||
intros.
|
|
||||||
transitivity (u px e).
|
|
||||||
apply transport_const.
|
|
||||||
apply nrP.
|
|
||||||
- cbn.
|
|
||||||
intros.
|
|
||||||
transitivity (u (l x0) (l x0)).
|
|
||||||
apply transport_const.
|
|
||||||
apply idemP.
|
|
||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
|
Set Implicit Arguments.
|
||||||
|
|
||||||
|
Definition comprehension :
|
||||||
Theorem FSet_rec_beta_assocT : forall
|
|
||||||
(A : Type)
|
|
||||||
(P : Type)
|
|
||||||
(e : P)
|
|
||||||
(l : A -> P)
|
|
||||||
(u : P -> P -> P)
|
|
||||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
|
||||||
(commP : forall (x y : P), u x y = u y x)
|
|
||||||
(nlP : forall (x : P), u e x = x)
|
|
||||||
(nrP : forall (x : P), u x e = x)
|
|
||||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
|
||||||
(x y z : FSet A),
|
|
||||||
ap (FSetRec e l u assocP commP nlP nrP idemP) (assoc x y z)
|
|
||||||
=
|
|
||||||
(assocP (FSetRec e l u assocP commP nlP nrP idemP x)
|
|
||||||
(FSetRec e l u assocP commP nlP nrP idemP y)
|
|
||||||
(FSetRec e l u assocP commP nlP nrP idemP z)
|
|
||||||
).
|
|
||||||
Proof.
|
|
||||||
intros.
|
|
||||||
eapply (cancelL (transport_const (assoc x y z) _ ) ).
|
|
||||||
simple refine
|
|
||||||
((apD_const
|
|
||||||
(FSetRec e l u assocP commP nlP nrP idemP)
|
|
||||||
(assoc x y z))^ @ _).
|
|
||||||
apply FSet_ind_beta_assoc.
|
|
||||||
|
|
||||||
|
|
||||||
Theorem FSet_rec_beta_commT : forall
|
|
||||||
(A : Type)
|
|
||||||
(P : Type)
|
|
||||||
(e : P)
|
|
||||||
(l : A -> P)
|
|
||||||
(u : P -> P -> P)
|
|
||||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
|
||||||
(commP : forall (x y : P), u x y = u y x)
|
|
||||||
(nlP : forall (x : P), u e x = x)
|
|
||||||
(nrP : forall (x : P), u x e = x)
|
|
||||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
|
||||||
(x y : FSet A),
|
|
||||||
ap (FSet_rec e l u assocP commP nlP nrP idemP) (comm x y)
|
|
||||||
=
|
|
||||||
(commP (FSet_rec e l u assocP commP nlP nrP idemP x)
|
|
||||||
(FSet_rec e l u assocP commP nlP nrP idemP y)
|
|
||||||
).
|
|
||||||
|
|
||||||
Axiom FSet_rec_beta_nl : forall
|
|
||||||
(A : Type)
|
|
||||||
(P : Type)
|
|
||||||
(e : P)
|
|
||||||
(l : A -> P)
|
|
||||||
(u : P -> P -> P)
|
|
||||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
|
||||||
(commP : forall (x y : P), u x y = u y x)
|
|
||||||
(nlP : forall (x : P), u e x = x)
|
|
||||||
(nrP : forall (x : P), u x e = x)
|
|
||||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
|
||||||
(x : FSet A),
|
|
||||||
ap (FSet_rec e l u assocP commP nlP nrP idemP) (nl x)
|
|
||||||
=
|
|
||||||
(nlP (FSet_rec e l u assocP commP nlP nrP idemP x)
|
|
||||||
).
|
|
||||||
|
|
||||||
Axiom FSet_rec_beta_nr : forall
|
|
||||||
(A : Type)
|
|
||||||
(P : Type)
|
|
||||||
(e : P)
|
|
||||||
(l : A -> P)
|
|
||||||
(u : P -> P -> P)
|
|
||||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
|
||||||
(commP : forall (x y : P), u x y = u y x)
|
|
||||||
(nlP : forall (x : P), u e x = x)
|
|
||||||
(nrP : forall (x : P), u x e = x)
|
|
||||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
|
||||||
(x : FSet A),
|
|
||||||
ap (FSet_rec e l u assocP commP nlP nrP idemP) (nr x)
|
|
||||||
=
|
|
||||||
(nrP (FSet_rec e l u assocP commP nlP nrP idemP x)
|
|
||||||
).
|
|
||||||
|
|
||||||
Axiom FSet_rec_beta_idem : forall
|
|
||||||
(A : Type)
|
|
||||||
(P : Type)
|
|
||||||
(e : P)
|
|
||||||
(l : A -> P)
|
|
||||||
(u : P -> P -> P)
|
|
||||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
|
||||||
(commP : forall (x y : P), u x y = u y x)
|
|
||||||
(nlP : forall (x : P), u e x = x)
|
|
||||||
(nrP : forall (x : P), u x e = x)
|
|
||||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
|
||||||
(x : A),
|
|
||||||
ap (FSet_rec e l u assocP commP nlP nrP idemP) (idem x)
|
|
||||||
=
|
|
||||||
idemP x.
|
|
||||||
|
|
||||||
|
|
||||||
End FinSet.
|
|
||||||
|
|
||||||
Definition isIn : forall
|
|
||||||
(A : Type)
|
|
||||||
(eq : A -> A -> Bool),
|
|
||||||
A -> FSet A -> Bool.
|
|
||||||
Proof.
|
|
||||||
intro A.
|
|
||||||
intro eq.
|
|
||||||
intro a.
|
|
||||||
refine (FSet_rec A _ _ _ _ _ _ _ _ _).
|
|
||||||
Unshelve.
|
|
||||||
|
|
||||||
Focus 6.
|
|
||||||
apply false.
|
|
||||||
|
|
||||||
Focus 6.
|
|
||||||
intro a'.
|
|
||||||
apply (eq a a').
|
|
||||||
|
|
||||||
Focus 6.
|
|
||||||
intro b.
|
|
||||||
intro b'.
|
|
||||||
apply (orb b b').
|
|
||||||
|
|
||||||
Focus 3.
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
reflexivity.
|
|
||||||
|
|
||||||
Focus 1.
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
destruct x.
|
|
||||||
reflexivity.
|
|
||||||
destruct y.
|
|
||||||
reflexivity.
|
|
||||||
reflexivity.
|
|
||||||
|
|
||||||
Focus 1.
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
destruct x.
|
|
||||||
destruct y.
|
|
||||||
reflexivity.
|
|
||||||
reflexivity.
|
|
||||||
destruct y.
|
|
||||||
reflexivity.
|
|
||||||
reflexivity.
|
|
||||||
|
|
||||||
Focus 1.
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
destruct x.
|
|
||||||
reflexivity.
|
|
||||||
reflexivity.
|
|
||||||
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
destruct (eq a x).
|
|
||||||
reflexivity.
|
|
||||||
reflexivity.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Definition comprehension : forall
|
|
||||||
(A : Type)
|
|
||||||
(eq : A -> A -> Bool),
|
|
||||||
(A -> Bool) -> FSet A -> FSet A.
|
(A -> Bool) -> FSet A -> FSet A.
|
||||||
Proof.
|
Proof.
|
||||||
intro A.
|
intros P.
|
||||||
intro eq.
|
simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
|
||||||
intro phi.
|
- apply E.
|
||||||
refine (FSet_rec A _ _ _ _ _ _ _ _ _).
|
- intro a.
|
||||||
Unshelve.
|
refine (if (P a) then L a else E).
|
||||||
|
- apply U.
|
||||||
Focus 6.
|
- intros. cbv. apply assoc.
|
||||||
apply empty.
|
- intros. cbv. apply comm.
|
||||||
|
- intros. cbv. apply nl.
|
||||||
Focus 6.
|
- intros. cbv. apply nr.
|
||||||
intro a.
|
- intros. cbv.
|
||||||
apply (if (phi a) then L A a else (empty A)).
|
destruct (P x); simpl.
|
||||||
|
+ apply idem.
|
||||||
Focus 6.
|
+ apply nl.
|
||||||
intro x.
|
|
||||||
intro y.
|
|
||||||
apply (U A x y).
|
|
||||||
|
|
||||||
Focus 3.
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
apply nl.
|
|
||||||
|
|
||||||
Focus 1.
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
apply assoc.
|
|
||||||
|
|
||||||
Focus 1.
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
apply comm.
|
|
||||||
|
|
||||||
Focus 1.
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
apply nr.
|
|
||||||
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
destruct (phi x).
|
|
||||||
apply idem.
|
|
||||||
apply nl.
|
|
||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
Definition intersection : forall (A : Type) (eq : A -> A -> Bool),
|
Definition intersection :
|
||||||
FSet A -> FSet A -> FSet A.
|
FSet A -> FSet A -> FSet A.
|
||||||
Proof.
|
Proof.
|
||||||
intro A.
|
intros X Y.
|
||||||
intro eq.
|
apply (comprehension (fun (a : A) => isIn a X) Y).
|
||||||
intro x.
|
|
||||||
intro y.
|
|
||||||
apply (comprehension A eq (fun (a : A) => isIn A eq a x) y).
|
|
||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
Definition subset (A : Type) (eq : A -> A -> Bool) (x : FSet A) (y : FSet A) : Bool.
|
Lemma intersection_E : forall x,
|
||||||
|
intersection E x = E.
|
||||||
Proof.
|
Proof.
|
||||||
refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
|
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
|
||||||
Unshelve.
|
- reflexivity.
|
||||||
|
- intro a.
|
||||||
Focus 6.
|
|
||||||
apply x.
|
|
||||||
|
|
||||||
Focus 6.
|
|
||||||
apply true.
|
|
||||||
|
|
||||||
Focus 6.
|
|
||||||
intro a.
|
|
||||||
apply (isIn A eq a y).
|
|
||||||
|
|
||||||
Focus 6.
|
|
||||||
intro b.
|
|
||||||
intro b'.
|
|
||||||
apply (andb b b').
|
|
||||||
|
|
||||||
Focus 1.
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
destruct x0.
|
|
||||||
destruct y0.
|
|
||||||
reflexivity.
|
|
||||||
reflexivity.
|
|
||||||
reflexivity.
|
|
||||||
|
|
||||||
Focus 1.
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
destruct x0.
|
|
||||||
destruct y0.
|
|
||||||
reflexivity.
|
|
||||||
reflexivity.
|
|
||||||
destruct y0.
|
|
||||||
reflexivity.
|
|
||||||
reflexivity.
|
|
||||||
|
|
||||||
Focus 1.
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
reflexivity.
|
|
||||||
|
|
||||||
Focus 1.
|
|
||||||
intros.
|
|
||||||
compute.
|
|
||||||
destruct x0.
|
|
||||||
reflexivity.
|
|
||||||
reflexivity.
|
|
||||||
|
|
||||||
intros.
|
|
||||||
destruct (isIn A eq x0 y).
|
|
||||||
compute.
|
|
||||||
reflexivity.
|
|
||||||
compute.
|
|
||||||
reflexivity.
|
reflexivity.
|
||||||
|
- unfold intersection.
|
||||||
|
intros x y P Q.
|
||||||
|
cbn.
|
||||||
|
rewrite P.
|
||||||
|
rewrite Q.
|
||||||
|
apply nl.
|
||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
Definition equal_set (A : Type) (eq : A -> A -> Bool) (x : FSet A) (y : FSet A) : Bool
|
Theorem intersection_La : forall a x,
|
||||||
:= andb (subset A eq x y) (subset A eq y x).
|
intersection (L a) x = if isIn a x then L a else E.
|
||||||
|
|
||||||
Fixpoint eq_nat n m : Bool :=
|
|
||||||
match n, m with
|
|
||||||
| O, O => true
|
|
||||||
| O, S _ => false
|
|
||||||
| S _, O => false
|
|
||||||
| S n1, S m1 => eq_nat n1 m1
|
|
||||||
end.
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
Theorem test : forall (A:Type) (eq : A -> A -> Bool)
|
|
||||||
(u: FSet A), ~(u = empty _) -> exists (a: A) (v: FSet A),
|
|
||||||
u = U _ (L _ a) v /\ (isIn _ eq a v) = False.
|
|
||||||
Proof.
|
Proof.
|
||||||
intros A eq.
|
intro a.
|
||||||
i
|
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
|
||||||
|
- reflexivity.
|
||||||
|
- intro b.
|
||||||
|
admit.
|
||||||
|
- unfold intersection.
|
||||||
|
intros x y P Q.
|
||||||
|
cbn.
|
||||||
|
rewrite P.
|
||||||
|
rewrite Q.
|
||||||
|
destruct (isIn a x) ; destruct (isIn a y).
|
||||||
|
* apply idem.
|
||||||
|
* apply nr.
|
||||||
|
* apply nl.
|
||||||
|
* apply nl.
|
||||||
|
Admitted.
|
||||||
|
|
||||||
|
Theorem comprehension_or : forall ϕ ψ x,
|
||||||
|
comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x) (comprehension ψ x).
|
||||||
|
Proof.
|
||||||
|
intros ϕ ψ.
|
||||||
|
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
|
||||||
|
- cbn. symmetry ; apply nl.
|
||||||
|
- cbn. intros.
|
||||||
|
destruct (ϕ a) ; destruct (ψ a) ; symmetry.
|
||||||
|
* apply idem.
|
||||||
|
* apply nr.
|
||||||
|
* apply nl.
|
||||||
|
* apply nl.
|
||||||
|
- simpl. intros x y P Q.
|
||||||
|
cbn.
|
||||||
|
rewrite P.
|
||||||
|
rewrite Q.
|
||||||
|
|
||||||
|
|
||||||
|
Theorem intersection_assoc : forall x y z,
|
||||||
|
intersection x (intersection y z) = intersection (intersection x y) z.
|
||||||
|
Proof.
|
||||||
|
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
|
||||||
|
- cbn.
|
||||||
|
intros y z.
|
||||||
|
rewrite intersection_E.
|
||||||
|
rewrite intersection_E.
|
||||||
|
rewrite intersection_E.
|
||||||
|
reflexivity.
|
||||||
|
- intro a.
|
||||||
|
cbn.
|
||||||
|
intros y z.
|
||||||
|
(* simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _ y) ; try (intros ; apply set_path2). *)
|
||||||
|
admit.
|
||||||
|
- unfold intersection.
|
||||||
|
intros x y P Q z z'.
|
||||||
|
cbn.
|
||||||
|
rewrite Q.
|
||||||
|
|
||||||
|
rewrite intersection_La.
|
||||||
|
rewrite intersection_La.
|
||||||
|
destruct (isIn a y).
|
||||||
|
* rewrite intersection_La.
|
||||||
|
destruct (isIn a (intersection y z)).
|
||||||
|
+ reflexivity.
|
||||||
|
+
|
||||||
|
*
|
||||||
|
destruct (isIn a (intersection y z)).
|
||||||
|
|
||||||
Loading…
Reference in New Issue